Last time, we learned to solve systems of linear equations both graphically and using the substitution method. Today we we will learn one more method of solving these problems, called the addition method (or sometimes the elimination method). It is based on the principle that, given two equations, we are allowed to add them together, that is, add the left sides and add the right sides, to make a new equation. The following video has two examples of this process, one simple and one more complicated.
Systems of equations can be used to solve many kinds of real-world applications. These problems can be very different from one another, and it is not always obvious how to create the correct equations based on the problem statement. Below are several such problems, together with videos explaining in detail how to solve them:
1. An electronics warehouse ships televisions and DVD players in certain combinations to retailers throughout the country. The weight of 3 televisions and 5 DVD players is 62.5 pounds, and the weight of 3 televisions and 2 DVD players is 52 pounds. Create a system of equations that represents this situation, and solve it to find out how much each television and DVD player weighs.
2. Jason bicycled from home to the train station at an average speed of 10 miles per hour. Then he boarded a train and traveled into the city at an average speed of 50 mph. The entire distance was 30 miles; the entire trip took 1 hour. How many miles did Jason travel by train?
3. Lisa will make punch that is 25% fruit juice by adding pure (100%) fruit juice to a 2-liter mixture that is 10% fruit juice. How many liters of pure fruit juice does she need to add? NOTE: the video should automatically begin playing at the start of the second problem, 2 minutes and 35 seconds after the start.
Here are the word problems that Prof Halleck gave in class:
A. At the carousel in DUMBO, five hot dogs and one drink cost $16. Two hot dogs and three drinks cost $9. Find the cost per hot dog and the cost per drink.
B. A plane flies 660 mi from Atlanta to Miami in 1.2 hours when traveling with a tailwind. The return flight against the same wind takes 1.5 hours. Find the speed of the plane in still air, and the speed of the wind.
C. Bob and Judy have decided to sell lemonade on the Brooklyn Bridge. Bob makes 4 gallons of lemonade that is 10% lemon juice. Judy thinks it’s too watery, so she is going to add some lemonade concentrate, which is 50% lemon juice. How many gallons of concentrate should she add to bring the mixture up to 25% lemon juice?